LONDON—U.K. house prices hit a fresh record high for a second straight month in April, fueled by a lack of property for sale at a time of healthy mortgage finance availability—a combination which threatens to keep pushing prices higher for some time to come.

The continued upward pressure on house prices remains one of the thorns in the coalition government's relationship. Junior partners the Liberal Democrats have proposed a revised tax on the most expensive homes in order boost the Treasury's coffers with the help from...